Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 55 / 1 / b / Solution

Taking the exterior derivative of the left-invariant differential forms yieldsFor example, $d({\rho }^{-1}d{x}^1)=-{\rho }^{-2}d\rho \wedge d{x}^1$. With the antisymmetric structure constants of a Lie algebra found above, the two terms with indices $(0,1)$ and $(1,0)$ combine asThe same computation holds for superscript $2$, and all superscript-$0$ terms vanish. Hence the Maurer-Cartan equation isThe factor $1/2$ avoids counting each antisymmetric pair twice. If coefficients were not required to be antisymmetric in $j,k$, their symmetric parts would be invisible to the wedge product of differential forms; the standard structure constants of a Lie algebra fix this ambiguity.